Optimal. Leaf size=94 \[ \frac {1}{3} \left (x^3-1\right )^{2/3} x+\frac {2}{9} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {195, 239} \begin {gather*} \frac {1}{3} \left (x^3-1\right )^{2/3} x+\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 239
Rubi steps
\begin {align*} \int \left (-1+x^3\right )^{2/3} \, dx &=\frac {1}{3} x \left (-1+x^3\right )^{2/3}-\frac {2}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=\frac {1}{3} x \left (-1+x^3\right )^{2/3}-\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 99, normalized size = 1.05 \begin {gather*} \frac {3 (x-1) \left (x^3-1\right )^{2/3} F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};-\frac {x-1}{1-(-1)^{2/3}},-\frac {x-1}{1+\sqrt [3]{-1}}\right )}{5 \left (\frac {x-1}{1+\sqrt [3]{-1}}+1\right )^{2/3} \left (\frac {x-1}{1-(-1)^{2/3}}+1\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.17, size = 94, normalized size = 1.00 \begin {gather*} \frac {1}{3} x \left (-1+x^3\right )^{2/3}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{9} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 86, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {2}{9} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{9} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} - 1\right )}^{\frac {2}{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.06, size = 30, normalized size = 0.32
method | result | size |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}}}\) | \(30\) |
risch | \(\frac {x \left (x^{3}-1\right )^{\frac {2}{3}}}{3}-\frac {2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(42\) |
trager | \(\frac {x \left (x^{3}-1\right )^{\frac {2}{3}}}{3}+\frac {2 \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-2 x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{9}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-2 x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{9}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 94, normalized size = 1.00 \begin {gather*} \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 26, normalized size = 0.28 \begin {gather*} \frac {x\,{\left (x^3-1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right )}{{\left (1-x^3\right )}^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.86, size = 32, normalized size = 0.34 \begin {gather*} - \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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